| L(s) = 1 | + (−0.190 + 0.587i)2-s + (0.363 + 2.29i)3-s + (1.30 + 0.951i)4-s + (−0.530 − 2.17i)5-s + (−1.41 − 0.224i)6-s + 2.34i·7-s + (−1.80 + 1.31i)8-s + (−2.27 + 0.739i)9-s + (1.37 + 0.103i)10-s + (−0.0249 − 0.0489i)11-s + (−1.70 + 3.34i)12-s + (−2.23 + 2.82i)13-s + (−1.37 − 0.447i)14-s + (4.78 − 2.00i)15-s + (0.572 + 1.76i)16-s + (4.43 + 0.702i)17-s + ⋯ |
| L(s) = 1 | + (−0.135 + 0.415i)2-s + (0.209 + 1.32i)3-s + (0.654 + 0.475i)4-s + (−0.237 − 0.971i)5-s + (−0.578 − 0.0916i)6-s + 0.886i·7-s + (−0.639 + 0.464i)8-s + (−0.758 + 0.246i)9-s + (0.435 + 0.0326i)10-s + (−0.00751 − 0.0147i)11-s + (−0.492 + 0.966i)12-s + (−0.619 + 0.784i)13-s + (−0.368 − 0.119i)14-s + (1.23 − 0.517i)15-s + (0.143 + 0.440i)16-s + (1.07 + 0.170i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.651 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.582804 + 1.26944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.582804 + 1.26944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.530 + 2.17i)T \) |
| 13 | \( 1 + (2.23 - 2.82i)T \) |
| good | 2 | \( 1 + (0.190 - 0.587i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.363 - 2.29i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 - 2.34iT - 7T^{2} \) |
| 11 | \( 1 + (0.0249 + 0.0489i)T + (-6.46 + 8.89i)T^{2} \) |
| 17 | \( 1 + (-4.43 - 0.702i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-0.535 + 3.38i)T + (-18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (2.13 + 4.19i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (-1.49 + 2.06i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.19 - 0.189i)T + (29.4 + 9.57i)T^{2} \) |
| 37 | \( 1 + (0.103 - 0.0335i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.21 + 2.37i)T + (-24.0 - 33.1i)T^{2} \) |
| 43 | \( 1 + (-3.16 + 3.16i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.69 + 6.45i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.13 - 13.4i)T + (-50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.55 + 6.98i)T + (-34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (2.50 - 7.71i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (-11.4 + 8.30i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (1.24 + 7.86i)T + (-67.5 + 21.9i)T^{2} \) |
| 73 | \( 1 + (2.56 - 7.88i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-5.27 + 7.25i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-3.52 - 4.85i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.837 - 0.426i)T + (52.3 - 72.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 8.32i)T + (29.9 + 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14238151253368502712109640602, −11.01259013766029841926667656352, −9.859263754093676040674552306952, −9.017799202437276713460682638625, −8.448018106286105960439937591972, −7.30872062837806503935853354702, −5.89550356228579554499783172390, −4.92423964402921756381926610977, −3.87356640407183986953740486595, −2.48703907173519036837576759668,
1.08029179597625863256144594285, 2.45433185370204295706983343722, 3.52565775190321141898340095563, 5.67738382142200387688958737086, 6.66751422645928612963889263255, 7.45750237752404848204830522887, 7.899054897620922033525739941105, 9.864374501636782550091322559984, 10.32797096401838798442109497894, 11.36608346939162146354198862613
